Weingarten integration over noncommutative homogeneous spaces

Abstract

195-224 (2017) Weingarten integration over noncommutative homogeneous spaces Teodor Banica AbstractWe discuss an extension of the Weingarten formula, to the case of noncommutative homogeneous spaces, under suitable "easiness" assumptions. The spaces that we consider are noncommutative algebraic manifolds, generalizing the spaces of type X = G/H ⊂ C N , with H ⊂ G ⊂ UN being subgroups of the unitary group, subject to certain uniformity conditions. We discuss various axiomatization issues, then we establish the Weingarten formula, and we derive some probabilistic consequences. Intégration de Weingarten sur les espaces homogènes non commutatifsRésumé On présente une extension de la formule d'intégration de Weingarten, pour les espaces homogènes non commutatifs, vérifiant des hypothèses « d'aisance » adéquates. Les espaces qu'on considère sont des variétés algebriques non commutatives, généralisant les espaces du type X = G/H ⊂ C N , avec H ⊂ G ⊂ UN étant des sous-groupes du groupe unitaire, vérifiant certaines conditions d'uniformité. On traite d'abord les questions d'axiomatisation, ensuite on établit la formule de Weingarten, et on finit avec quelques conséquences probabilistes.

Tannakian Duality for Affine Homogeneous Spaces

“…We will answer here this question, with a Tannakian characterization of such manifolds. We believe that some further improvements of this result can lead to an axiomatization of the "easy algebraic manifolds", which was the main question in [1], and which remains open.…”

“…Here both the algebras on the right are by definition universal C * -algebras. Following [1], we can now formulate the following definition:…”

“…See [1]. We will need one more general result from [1], namely an extension of the Weingarten integration formula [2], [5], [6], [12], to the affine homogeneous space setting: Proposition 1.5. Assuming that G → X is an affine homogeneous space, with index set I ⊂ {1, .…”

“…The present work is a continuation of [1], which was concerned with the integration theory over the associated homogeneous spaces. The main finding there was the fact that, in order to have a good integration theory, one must restrict the attention to a certain special class of homogeneous spaces, called "affine".…”

“…One question left open in [1] was that of finding an abstract axiomatization of the algebraic manifolds X ⊂ S N −1 C,+ which can appear in this way. We will answer here this question, with a Tannakian characterization of such manifolds.…”